Having taken an early lead through new record signing Alexandre Lacazette, the Gunners found themselves 2-1 and then 3-2 down before coming back to snatch a 4-3 victory.
Cech told Sky Sports: "We started really well, we got the first goal and I thought it was the moment where we could control the game but then we made mistakes and they punished us.
"They had two half-chances and put them in and we had to chase again and I thought going in at 2-2 at half-time it was our advantage, but then the corner gave them the advantage again and we had to chase again.
"They were difficult moments when we were 2-3 down but we managed to get out of that and found two goals, which was great to see.
"We made some technical mistakes, for example the second goal, but that can happen, that's part of football. They put a great ball in and Jamie Vardy does what he does best.
"Overall we showed great character. First game of the season we played a classic, so for everybody at home it will have been amazing to watch."
Olivier Giroud, linked with a move away from the Emirates as he looks for more game time, scored the winning goal for Arsene Wenger's side after coming off the bench and he hopes the Gunners can now kick on.
He said: "I knew I had the chance of coming on in the game. We tried to push, we kept believing in our game, carried on, pushing hard and we've been courageous to the end and thank goodness we won this game.
"That's a nice start but obviously I prefer to be on the pitch.
"But as much as I can I will try to help the team to reach our target and tonight we needed this win as everybody knows the importance of the first game for confidence and we keep going.
"I'm very pleased for the team and hope we can carry on."